dominated convergence theorem

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21: 18.2 General Orthogonal Polynomials

… ► … ►Markov’s theorem states that … ►This says roughly that the series (18.2.25) has the same pointwise convergence behavior as the same series with $p_n(x)=T_n\left(x\right)$, a Chebyshev polynomial of the first kind, see Table 18.3.1. … ►Part of this theorem was already proved by Blumenthal (1898). … ►See Szegő (1975, Theorem 7.2). …

22: 17.18 Methods of Computation

… ►Method (1) is applicable within the circles of convergence of the defining series, although it is often cumbersome owing to slowness of convergence and/or severe cancellation. … ►Method (1) can sometimes be improved by application of convergence acceleration procedures; see §3.9. …

23: 10.74 Methods of Computation

… ►In other circumstances the power series are prone to slow convergence and heavy numerical cancellation. … ►Newton’s rule is quadratically convergent and Halley’s rule is cubically convergent. … ►To ensure that no zeros are overlooked, standard tools are the phase principle and Rouché’s theorem; see §1.10(iv). …

24: 30.10 Series and Integrals

… ►For an addition theorem, see Meixner and Schäfke (1954, p. 300) and King and Van Buren (1973). …

25: 10.44 Sums

… ►

§10.44(i) Multiplication Theorem

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§10.44(ii) Addition Theorems

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Neumann’s Addition Theorem

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Graf’s and Gegenbauer’s Addition Theorems

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26: 18.39 Applications in the Physical Sciences

… ►An important, and perhaps unexpected, feature of the EOP’s is now pointed out by noting that for 1D Schrödinger operators, or equivalent Sturm-Liouville ODEs, having discrete spectra with $L^2$ eigenfunctions vanishing at the end points, in this case $\pm \mathrm{\infty }$ see Simon (2005c, Theorem 3.3, p. 35), such eigenfunctions satisfy the Sturm oscillation theorem. …Both satisfy Sturm’s theorem. … ►

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Figure 18.39.1: Graphs of the first and fourth excited state eigenfunctions of the harmonic oscillator, for $\mathrm{\hslash }=k=m=1$, of (18.39.13), in ${\psi }_1(x)$, ${\psi }_4(x)$ and those of the rational potential of (18.39.19), in ${\widehat{\psi }}_3(x)$, ${\widehat{\psi }}_6(x)$. Both sets satisfy the Sturm oscillation theorem. Magnify

… ►Interactions between electrons, in many electron atoms, breaks this degeneracy as a function of $l$, but $n$ still dominates. …

27: 1.5 Calculus of Two or More Variables

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Implicit Function Theorem

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§1.5(iii) Taylor’s Theorem; Maxima and Minima

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§1.5(iv) Leibniz’s Theorem for Differentiation of Integrals

… ►Suppose also that ${\int }_c^{d}f(x,y)dy$ converges and ${\int }_c^d(\partial f/\partial x)dy$ converges uniformly on $a\le x\le b$, that is, given any positive number $ϵ$, however small, we can find a number $c_0\in [c,d)$ that is independent of $x$ and is such that … ►whenever both repeated integrals exist and at least one is absolutely convergent. …

28: 15.2 Definitions and Analytical Properties

… ►On the circle of convergence, $|z|=1$, the Gauss series: ►

Converges absolutely when $\mathrm{\Re }\left(c-a-b\right)>0$.

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Converges conditionally when and $z=1$ is excluded.

…

29: 16.2 Definition and Analytic Properties

… ►When $p\le q$ the series (16.2.1) converges for all finite values of $z$ and defines an entire function. … ►If none of the $a_j$ is a nonpositive integer, then the radius of convergence of the series (16.2.1) is $1$, and outside the open disk the generalized hypergeometric function is defined by analytic continuation with respect to $z$. … ►On the circle $|z|=1$ the series (16.2.1) is absolutely convergent if $\mathrm{\Re }{\gamma }_q>0$, convergent except at $z=1$ if , and divergent if $\mathrm{\Re }{\gamma }_q\le -1$, where …

30: 18.33 Polynomials Orthogonal on the Unit Circle

… ►

18.33.23 $${\mathrm{\Phi }}_{n+1}(z)=z{\mathrm{\Phi }}_n(z)-\overline{{\alpha }_n}{\mathrm{\Phi }}_n^{\ast }(z),$$

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Symbols:

$\overline{z}$: complex conjugate, $z$: complex variable and $n$: nonnegative integer

Proved:

Simon (2005a, Theorem 1.5.2)(proved)

Referenced by:

§18.33(vi), §18.33(vi), §18.33(vi), §18.33(vi)

Permalink:

http://dlmf.nist.gov/18.33.E23

Encodings:

pMML, png, TeX

See also:

Annotations for §18.33(vi), §18.33(vi), §18.33 and Ch.18

… ►

Verblunsky’s Theorem

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Szegő’s Theorem

►For $w(z)$ as in (18.33.19) (or more generally as the weight function of the absolutely continuous part of the measure $\mu$ in (18.33.17)) and with ${\alpha }_n$ the Verblunsky coefficients in (18.33.23), (18.33.24), Szegő’s theorem states that …By (18.33.25) , so the infinite product in (18.33.31) converges, although the limit may be zero. …